$K$-continuity is equivalent to $K$-exactness

TL;DR

This paper establishes that in $K$-theory, the property of $K$-continuity is equivalent to $K$-exactness, paralleling a known result in $C^*$-algebra tensor products, by leveraging Dadarlat's work.

Contribution

It proves the equivalence of $K$-continuity and $K$-exactness, providing a $K$-theoretic analogue to a classical tensor product result.

Findings

$K$-continuity is equivalent to $K$-exactness.

Uses Dadarlat's result to establish the equivalence.

Extends classical tensor product theory to $K$-theory.

Abstract

It is well known that the functor of taking the minimal tensor product with a fixed $C^*$-algebra preserves inductive limits if and only if it preserves extensions. In other words, tensor continuity is equivalent to tensor exactness. We consider a $K$-theoretic analogue of this result and show that $K$-continuity is equivalent to $K$-exactness, using a result of M. Dadarlat.